This function uses a method to determine the minimum sufficient embedding dimension. It is based on the False Nearest Neighbors section of the TISEAN documentation. As a demonstration we will create a plot that contains an Ikeda Map, a Henon Map and a Henon Map corrupted by 10% of Gaussian noise.

The dim_* variables hold the dimension (so here 1:5), and frac_* contain the fraction of false nearest neighbors. From this chart we can obtain the sufficient embedding dimension for each system. For a Henon Map m = 2 is sufficient, but for an Ikeda map it is better to use m = 3.

Here I will demonstrate how to find unstable periodic orbits. This section is based on the TISEAN documentation chapter Finding unstable periodic orbits. We will start by finding these orbits using function upo.

In this section we will demonstrate some functions from the 'Nonlinear Prediction' chapter of the TISEAN documentation (located here). For now this section will only demonstrate functions that are connected to the Simple Nonlinear Prediction section.
There are three functions in this section: lzo_test, lzo_gm and lzo_run. The first is used to estimate the forecast error for a set of chosen parameters, the second gives some global information about the fit and the third produces predicted points. Let us start with the first one (before starting this example remember to download 'amplitude.dat' from above and start Octave in the directory that contains it). The pairs of parameters (m,d) where chosen after the TISEAN documentation.

It seems that the last pair (m = 4, d = 6) is suitable. We will use it to determine the the best neighborhood to use when generating future points.

Code: Determining the best neighborhood size using lzo_gm

gm=lzo_gm(amplitude(1:end-200),'m',4,'d',6,'s',steps,'rhigh',20);

After analyzing gm it is easy to observe that the least error is for the first neighborhood size of gm which is equal to 0.49148. We will use it to produce the prediction vectors. Only the first 4800 datapoints from amplitude are used in order to compare the predicted values with the actual ones for the last 200 points.

This tutorial show different methods of the 'Nonlinear Noise Reduction' section of the TISEAN documentation (located here). It shows the use of simple nonlinear noise reduction (function lazy) and locally projective nonlinear noise reduction (function ghkss). To start let's create noisy data to work with.

Code: Creating a noisy henon map

hen=henon(10000);hen=hen(:,1);# We only need the first columnhen_noisy=hen+std(hen)*0.02.*(-6+sum(rand([size(hen),12]),3));

This created a Henon map contaminated by 2% Gaussian noise à la TISEAN. In the tutorials and exercises on the TISEAN website this would be equivalent to calling makenoise -%2 on the Henon map.
Next we will reduce the noise using simple nonlinear noise reduction lazy.

Code: Simple nonlinear noise reduction

clean=lazy(hen_noisy,7,-0.06,3);# Create delay vectors for both the clean and noisy datadelay_clean=delay(clean);delay_noisy=delay(hen_noisy);# Plot both on one chartplot(delay_noisy(:,1),delay_noisy(:,2),'b.;Noisy Data;','markersize',3,...delay_clean(:,1),delay_clean(:,2),'r.;Clean Data;','markersize',3)

On the chart created the red dots represent cleaned up data. It is much closer to the original than the noisy blue set.
Now we will do the same with ghkss.

Code: Locally projective nonlinear noise reduction

clean=ghkss(hen,'m',7,'q',2,'r',0.05,'k',20,'i',2);

The rest of the code is the same as the code used in the lazy example.
Once both results are compared it is quite obvious that for this particular example ghkss is superior to lazy. The TISEAN documentation points out that this is not always the case.

Here I will demonstrate how to use the function lyap_k. It estimates the maximal Lyapunov exponent from a time series (more information available from the TISEAN documentation located here). In this tutorial we will estimate the maximal Lyapunov exponent for various embedding dimensions and then plot them.

This section is discussed on the TISEAN documentation page. One of the functions discussed is d2. It is used to estimate the correlation sum, correlation dimension and correlation entropy of a time series. The time series used here will be the Henon map.

The output of d2 can be further processed using the following functions: av_d2, c2t, c2g. This tutorial will show how to use av_d2 which smooths the output of d2 (usually used to smooth the "d2" field of the output).

This section is discussed on the TISEAN documentation page. The focus of this section will be the function surrogates. It uses surrogate data to determine weather data is nonlinear. Let us first create the input data which will be a stationary Gaussian linear stochastic process. It is measured by s(xn) = xn^3. We then run it through surrogates and plot the data.

It is crucial that the length of the input to surrogates is factorizable by only 2,3 and 5. Therefore, if it is not the excess of data is truncated accordingly. Padding with zeros is not allowed. To solve this problem one can use endtoend, and choose the best subset of the input data to be used to generate a surrogate.